Recently, a number of spherical packing phases belonging to the class of Frank-Kasper (F-K) phases have been observed in a wide range of soft matter systems, including block copolymers, ionic surfactants, liquid crystalline dendrimers, and giant surfactants. Although their emergence in such systems has been conjectured to be due to a competition between mesodomain sphericity and incompressibility, we lack a description of a precise and general mechanism underlying the formation of F-K phases in soft matter systems. In this work, we consider the two most common F-K phases found in soft matter systems, the σ and A15 phases, and study their stability in the context of a well-known Landau model known as the Landau-Brazovskii model. This model has been applied to systems ranging from block copolymers to liquid crystals. We find that the phase behavior of the Landau-Brazovskii model is controlled only by two parameters, rather than by three parameters, as was suggested by previous works. We also find that the Landau-Brazovksii phase diagram contains regions in which the σ or A15 phase is the most stable among a set of candidate phases. The fact that such a simple model can predict these complex phases provides some insight into the question of why the occurrence of the Frank-Kasper phases in soft matter is so widespread. / Thesis / Master of Science (MSc) / Soft condensed matter physics studies the properties of materials that easily deform, such as soap, gel and plastic. Many of these materials can self-assemble into various fascinating ordered structures. One particularly complex class of structures, found in a wide range of soft materials, is the class of Frank-Kasper phases. Frank-Kasper phases in soft materials have potential applications in fields such as photonics, so their formation in these materials is particularly interesting. However, it is not well understood why the Frank-Kasper structures occur in so many soft materials. We investigate this problem, and show that the occurrence of these structures might be described by a very simple mathematical model known as the Landau-Brazovskii model. The fact that such a simple model can predict the complex Frank-Kasper phases provides insight into the origin of the widespread nature of the occurrence of Frank-Kasper phases in soft materials.

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Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: Two problems in the field of materials-based condensed matter physics, specifically in the field of superconductivity, are studied theoretically. In both problems, where each is of current exper- imental interest, an extension of Ginzburg-Landau theory is used to describe a physical system, with focus on the energy associated to the interface(s) occurring in the respective systems. The first physical system under consideration is that of a two-band superconductor. Using Ginzburg-Landau theory for two-band superconductors, the interface energy ¾s between normal and superconducting states coexisting at the thermodynamic critical magnetic field is determined. From the theoretical and numerical analysis of the interface energy, it is found that close to the transition temperature, where the Ginzburg-Landau theory is applicable, the two-band problem maps onto an effective single band problem. This finding puts into question the possibility of intermediate, so called type-1.5 superconductivity, in the regime where the Ginzburg-Landau theory applies. The second physical system is that of a system with competing superconductivity and anti- ferromagnetism. From Ginzburg-Landau theory for such competing systems in a thermodynamic critical magnetic field, it is shown that two possible interfaces can occur: an interface between a pure anti-ferromagnetic state and a pure superconducting state; and an interface between a state with coexisting superconductivity and anti-ferromagnetism and a pure anti-ferromagnetic state. The energy associated to both these interfaces is analysed theoretically and numerically from which the boundary between type-I and type-II superconductivity is obtained for certain specific cases. / AFRIKAANSE OPSOMMING: Twee probleme in die veld van materiaal-gebaseerde gekondenseerde materie fisika, spesifiek in die veld van supergeleiding, word teoreties bestudeer. In beide probleme, albei tans van eksper- imentele belang, word ’n fisiese sisteem beskryf deur ’n uitbreiding van enkel-band Ginzburg- Landau teorie, met fokus op die energie geassosieer met die koppelvlak(ke) wat in die onderskeie sisteme aangetref word. Die eerste fisiese sisteem wat beskou word is die van ’n twee-band supergeleier. Deur van Ginzburg-Landau teorie vir twee-band supergeleiers gebruik te maak, word die koppelvlak energie ¾s tussen die gelyktydig bestaande normaal- en supergeleidende toestand in die termodinamiese kritieke magneetveld bepaal. Deur beide teoretiese en numeriese analieses word bepaal dat na aan die oorgangstemperatuur, waar Ginzburg-Landau teorie geldig is, die twee-band probleem op ’n effektiewe een-band probleem afbeeld. Hierdie bevinding bevraagteken dus die moontlikheid van onkonvensionele, of sogenaamde tipe-1.5 supergeleiding, vir gevalle waar Ginzburg-Landau teorie geldig is. Die tweede fisiese siteem wat beskou word is ’n sisteem met kompeterende supergeleiding en anti-ferromagnetisme. Met behulp van Ginzburg-Landau teorie vir sulke sisteme in ’n termod- inamiese kritiese magneetveld word gewys dat daar twee moontlike koppelvlakke kan ontstaan: ’n koppelvlak tussen ’n uitsluitlik anti-ferromagnetiese toestand en ’n uitsluitlik supergeleidende toestand; sowel as ’n koppelvlak tussen ’n uitsluitlik anti-ferromagnetiese toestand en ’n toes- tand van beide supergeleiding en anti-ferromagnetisme. Die energie geassosieer met beide hierdie koppelvlakke word teoreties en numeries geanaliseer wat lei tot ’n beskrywing van die grenslyn tussen tipe-I en tipe-II supergeleiding in sekere spesifieke gevalle.

<p>Chapter 1 to 4 give a short introduction to superconductivity, microscopic theory, phase transitions, and Monte-Carlo simulations. Chapter 2 is about Cooper pairing in different settings, but I also give a short introduction to the Hofstadter problem of lattice fermions on a square lattice in a perpendicular magnetic field. The purpose is to clarify some points in Paper-I. Chapter 3 is about phase transitions, and introduces the important concepts of spontaneous symmetry breaking, scaling, and renormalization. In the last section I stress some of the main differences between first order and second order phase transitions. Chapter 4 starts with a short elementary introduction to Monte-Carlo simulations and proceeds with the important, but somewhat more advanced topic of reweighting.</p><p>Chapter 5 to 7 are more closely related to the specific projects I have worked on, and are meant to illuminate and clarify some aspects in Paper-II and Paper-III. Chapter 5 introduce the Ginzburg-Landau model in various parametrizations, present some perturbative (mean-field) results, and introduce the concept of topological defects (vortices) and duality.</p><p>Chapter 6 is closely related to Paper-II and introduce the concept of fractal dimension and the relation between the vortex excitations of the original theory and the dual field theory. Chapter 7 is closely related to Paper-III where we studied the order of the metal to superconductor phase transition. To do this we had to do infinite volume and continuum limit extrapolations. We also had to consider ultraviolet renormalization since the Ginzburg-Landau theory is a continuum field theory with no inherent short scale cut-off. To reduce auto-correlation times we added several improvements to the standard Metropolis algorithm in the Monte-Carlo simulations, the most important being an overrelaxation algorithm for the scalar field and a global update of the scalar amplitude.</p>

Chapter 1 to 4 give a short introduction to superconductivity, microscopic theory, phase transitions, and Monte-Carlo simulations. Chapter 2 is about Cooper pairing in different settings, but I also give a short introduction to the Hofstadter problem of lattice fermions on a square lattice in a perpendicular magnetic field. The purpose is to clarify some points in Paper-I. Chapter 3 is about phase transitions, and introduces the important concepts of spontaneous symmetry breaking, scaling, and renormalization. In the last section I stress some of the main differences between first order and second order phase transitions. Chapter 4 starts with a short elementary introduction to Monte-Carlo simulations and proceeds with the important, but somewhat more advanced topic of reweighting. Chapter 5 to 7 are more closely related to the specific projects I have worked on, and are meant to illuminate and clarify some aspects in Paper-II and Paper-III. Chapter 5 introduce the Ginzburg-Landau model in various parametrizations, present some perturbative (mean-field) results, and introduce the concept of topological defects (vortices) and duality. Chapter 6 is closely related to Paper-II and introduce the concept of fractal dimension and the relation between the vortex excitations of the original theory and the dual field theory. Chapter 7 is closely related to Paper-III where we studied the order of the metal to superconductor phase transition. To do this we had to do infinite volume and continuum limit extrapolations. We also had to consider ultraviolet renormalization since the Ginzburg-Landau theory is a continuum field theory with no inherent short scale cut-off. To reduce auto-correlation times we added several improvements to the standard Metropolis algorithm in the Monte-Carlo simulations, the most important being an overrelaxation algorithm for the scalar field and a global update of the scalar amplitude.

Nós estudamos a competição entre a instabilidade de Pomeranchuk no canal de spin com momento angular l=1 e uma interação atrativa, favorecendo a formação de um par de Cooper. Achamos, numa aproximação de campo médio, uma forte supressão da instabilidade de Pomeranchuk via supercondutividade. Além disso, identificamos uma fase supercondutora metaestável com características semelhantes ao estado FFLO. Um líquido de Fermi é, com exceção de uma dimensão, um estado muito estável da matéria. Por outro lado dois tipos de instabilidades, relacionadas com interações atrativas, são conhecidas: Instabilidades Pomeranchuk e supercondutora. As instabilidades Pomeranchuk ocorrem na presença da interação de dois corpos contendo uma forte componente atrativa no canal de espalhamento para frente com momento angular definido. No contexto da teoria de Landau, a instabilidade ocorre quando um ou mais parâmetros admensionais de Landau nos canais de spin ou carga, adquirem altos valores negativos. As instabilidades Pomeranchuk no setor de carga quebram a simetria de rotação. Em particular, uma instabilidade em alguns canais produz uma deformação elipsoidal na superfície de Fermi. / We study the competition between a Pomeranchuk instability in the spin channel with angular momentum l = 1 and an attractive interaction, favoring Cooper pair formation. We find, at mean-field approximation, that superconductivity strongly suppress the Pomeranchuk instability. Moreover, we have found a metastable modulated superconducting phase with similar characteristics of the FFLO state. A Fermi liquid is, except in one dimension, a very stable state of matter. At least two types of instabilities, related with attractive interactions, are known: Pomeranchuk and superconducting instabilities. Pomeranchuk instabilities occur in the presence of two-body interactions containing a strong attractive component in the forward scattering channel with a definite angular momentum. In the context of Landau theory, the instability sets in when one or more dimensionless Landau parameters in the charge or spin channel, acquire large negative values. Pomeranchuk instabilities in a charge sector spontaneously break rotational symmetry. In particular, an instability in the some channels produces an ellipsoidal deformation of the Fermi surface.

Nós estudamos a competição entre a instabilidade de Pomeranchuk no canal de spin com momento angular l=1 e uma interação atrativa, favorecendo a formação de um par de Cooper. Achamos, numa aproximação de campo médio, uma forte supressão da instabilidade de Pomeranchuk via supercondutividade. Além disso, identificamos uma fase supercondutora metaestável com características semelhantes ao estado FFLO. Um líquido de Fermi é, com exceção de uma dimensão, um estado muito estável da matéria. Por outro lado dois tipos de instabilidades, relacionadas com interações atrativas, são conhecidas: Instabilidades Pomeranchuk e supercondutora. As instabilidades Pomeranchuk ocorrem na presença da interação de dois corpos contendo uma forte componente atrativa no canal de espalhamento para frente com momento angular definido. No contexto da teoria de Landau, a instabilidade ocorre quando um ou mais parâmetros admensionais de Landau nos canais de spin ou carga, adquirem altos valores negativos. As instabilidades Pomeranchuk no setor de carga quebram a simetria de rotação. Em particular, uma instabilidade em alguns canais produz uma deformação elipsoidal na superfície de Fermi. / We study the competition between a Pomeranchuk instability in the spin channel with angular momentum l = 1 and an attractive interaction, favoring Cooper pair formation. We find, at mean-field approximation, that superconductivity strongly suppress the Pomeranchuk instability. Moreover, we have found a metastable modulated superconducting phase with similar characteristics of the FFLO state. A Fermi liquid is, except in one dimension, a very stable state of matter. At least two types of instabilities, related with attractive interactions, are known: Pomeranchuk and superconducting instabilities. Pomeranchuk instabilities occur in the presence of two-body interactions containing a strong attractive component in the forward scattering channel with a definite angular momentum. In the context of Landau theory, the instability sets in when one or more dimensionless Landau parameters in the charge or spin channel, acquire large negative values. Pomeranchuk instabilities in a charge sector spontaneously break rotational symmetry. In particular, an instability in the some channels produces an ellipsoidal deformation of the Fermi surface.

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